D9d E 2 C9 2 C9^2 2 C3 2 C9^4 9 C2' i 2 S18 2 S6 2 S18^5 2 S18^7 9 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ... ....T ....... ........T ........... ............T A2g 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..T ... ..... ....... ......... ........... ............. E1g 2.0000 1.5320 0.3473 -1.0000 -1.8793 0.0000 2.0000 -1.8793 -1.0000 0.3473 1.5320 0.0000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.0000 0.3473 -1.8793 -1.0000 1.5320 0.0000 2.0000 1.5320 -1.0000 -1.8793 0.3473 0.0000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.0000 -1.0000 -1.0000 2.0000 -1.0000 0.0000 2.0000 -1.0000 2.0000 -1.0000 -1.0000 0.0000 ... ... ..... ....... ..TT..... ........... TT....TT..... E4g 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.0000 2.0000 0.3473 -1.0000 1.5320 -1.8793 0.0000 ... ... ..... ....... TT....... ........... ..TTTT....... A1u 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. A2u 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 1.0000 ... ..T ..... ......T ......... ..........T ............. E1u 2.0000 1.5320 0.3473 -1.0000 -1.8793 0.0000 -2.0000 1.8793 1.0000 -0.3473 -1.5320 0.0000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.0000 0.3473 -1.8793 -1.0000 1.5320 0.0000 -2.0000 -1.5320 1.0000 1.8793 -0.3473 0.0000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.0000 -1.0000 -1.0000 2.0000 -1.0000 0.0000 -2.0000 1.0000 -2.0000 1.0000 1.0000 0.0000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.0000 -2.0000 -0.3473 1.0000 -1.5320 1.8793 0.0000 ... ... ..... ....... ......... TTTT....... ............. Irrational character values: 1.879385241572 = 2*cos(2*π/18) = 2*cos(π/9) 1.532088886238 = 2*cos(4*π/18) = 2*cos(2*π/9) 0.347296355334 = 2*cos(8*π/18) = 2*cos(4*π/9) Symmetry of Rotations and Cartesian products A1g d+g+i+k+2m z2, z4, z6 A2g R+m Rz E1g R+d+g+i+2k+3m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2g d+g+i+2k+2m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3g g+2i+2k+2m {xz(x2−3y2), yz(3x2−y2)}, {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4g g+2i+2k+2m {(x2−y2)2−4x2y2, xy(x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} A1u l A2u p+f+h+j+2l z, z3, z5 E1u p+f+h+j+2l {x, y}, {xz2, yz2}, {xz4, yz4} E2u f+h+2j+2l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3u f+h+2j+2l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4u 2h+2j+2l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)} Notes: α The order of the D9d point group is 36, and the order of the principal axis (S18) is 18. The group has 12 irreducible representations. β The D9d point group is isomorphic to D9h, C18v and D18. γ The D9d point group is generated by two symmetry elements, S18 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). The canonical choice, however, is to use redundant generators: C9, C2′ and i. δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Reversely, the single set of symmetry planes denoted σd contains the yz plane but not the xz plane. ε The lowest nonvanishing multipole moment in D9d is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. ι The regular nonagon or enneagon is not constructible by ruler and compass because cos(2*π/9) has an algebraic degree of 3. (It can be constructed by extended methods that allow angle trisection, as this corresponds to solving cubic equations). The value of cos(2*π/9) can be expressed using cubic roots and complex numbers, which, however, is not very useful for a real-valued quantity: 2*cos(2π/9) = (3√−4+i*4*√3 + 3√−4−i*4*√3)/2.
| D7d | ||
| D8d | ||
| C9 C9v C9h D9 D9h | D9d | |
| D10d | ||
| D11d |
This Character Table for the D9d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.